Most of the lattices \(\Gamma \)  as in Section REF  are finitely presented (see
Section REF ). The question whether \(\Gamma \)  is Kähler or not has been studied by Simpson
using his non-abelian Hodge theory when at least one of the \(k_i\) 's is
archimedean. He shows that if
\(\Gamma \)  is Kähler then necessarily for any \(i \in I\)  such
that \(k_i\)  is archimedean the group \({\mathbf {G}}_i\)  has to be of Hodge
type (i.e. admits a Cartan involution which is an inner automorphism),
see [1]}. For
example \({\mathbf {SL}}(n, {\mathbb {Z}})\)  is not a Kähler group
as \({\mathbf {SL}}(n, {\mathbb {R}})\)  is not a group of Hodge type. In this note we prove:
